For a project of mine I'm creating large (240mm tall) 7 segment displays using a lot of LEDs (245 per digit, 35 per segment).

I'm trying to work out if the ULN2803 will be enough to power each digit (datasheet: http://www.datasheetcatalog.org/datasheets/105/366825_DS.pdf ).

looking at the datasheet, it says the maximum current per channel is 500mA, well above the 140 mA (a 7 x 5 array of 20 mA LEDs) I will need per output, but the chip has a maximum power dissipation of 2.25 W (1W per channel)

So, am I correct in thinking that at the current I require the Vce will be 1.2V, so the power dissipated per channel will be (0.140 A * 1.2 V) 168 mW, putting my total at 1.18 W, and so the chip should meet my requirements?

Finally, anyone have any idea how hot the ship will actually get in practice at these requirements? Will I need to heat sink it?

Your numbers look right, but the chip will probably be getting pretty hot. The key number is in the datasheet page 3 under Thermal Data: thermal resistance junction-to-ambient of 55 C/W. As a first approximation, if your driver is dissipating 1.18W then it will heat up to 55*1.18=65 degrees above ambient (say, 25C) so a total junction temperature of 90C.

That's still in the safe area, but it will feel VERY hot to the touch.

"So, am I correct in thinking that at the current I require the Vce will be 1.2V, so the power dissipated per channel will be (0.140 A * 1.2 V) 168 mW, putting my total at 1.18 W, and so the chip should meet my requirements?"Your math and assumptions look good to me.

I would heatsink it - it will be less effective as a driver while it is hot.